Nomenclature

Input parameters

Thermodynamic variables

Turbine design parameters

Outputs

The flow coefficient ϕ=vm/u3\phi=v_{m}/u_3 is computed from the stage equations

The flow angles β3\beta_3 and β4\beta_4 are computed from the stage equations

The spouting velocity is given by

v0=2(h01h4s) v_0 = \sqrt{2(h_{01} - h_{4s})}

The blade velocity at each station is given by

u1=0u2=0u3=(r3/r4)u4=νv0\begin{gather} u_1 = 0 \\ u_2 = 0 \\ u_3 = (r_3/r_4) \\ u_4 = \nu \, v_0 \end{gather}

The constant meridional velocity is given by:

vm=ϕu4 v_m = \phi \, u_4

The absolute velocities are given by:

v1=vm/cos(α1)v2=vm/cos(α2)v3=vm/cos(α3)v4=vm/cos(α4)\begin{gather} v_1 = v_m / \cos(\alpha_1) \\ v_2 = v_m / \cos(\alpha_2) \\ v_3 = v_m / \cos(\alpha_3) \\ v_4 = v_m / \cos(\alpha_4) \\ \end{gather}

The relative velocities are given by:

w1=vm/cos(β1)w2=vm/cos(β2)w3=vm/cos(β3)w4=vm/cos(β4)\begin{gather} w_1 = v_m / \cos(\beta_1) \\ w_2 = v_m / \cos(\beta_2) \\ w_3 = v_m / \cos(\beta_3) \\ w_4 = v_m / \cos(\beta_4) \\ \end{gather}

The enthalpy of each station is given as

h1=h01v12/2 h_1 = h_{01} - v_{1}^2/2

The enthalpy at the exit of the stator os obtained from the equation for the conservation of energy:

h1h2u32=12ϕ2(tan2α2tan2α1)\frac{h_1 - h_2}{u_3^2} = \frac{1}{2} \phi^2 \left( \tan^2 \alpha_2 - \tan^2 \alpha_1 \right)

In the absence of losses and for constant meridional velocity, the enthalpy does not change across the interspace:

h3=h2,α3=α2h_3 = h_2, \quad \alpha_3 = \alpha_2

The enthalpy at the exit of the rotor is obtained from the conservation of rothalpy:

h3h4u32=12ϕ2(tan2β4tan2β3)12(1ρ2)\frac{h_3 - h_4}{u_3^2} = \frac{1}{2} \phi^2 \left( \tan^2 \beta_4 - \tan^2 \beta_3 \right)- \frac{1}{2}(1-\rho^2)

Once the enthalpy at each flow station is known, the thermodynamic state can be calculated assuming an isentropic process accross the turbine

[p1,ρ1,a1,q1]=EOS(h1,s1)[p2,ρ2,a2,q2]=EOS(h2,s2)[p3,ρ3,a3,q3]=EOS(h3,s3)[p4,ρ4,a4,q4]=EOS(h4,s4)\begin{gather} [p_1, \rho_1, a_1, q_1] = \mathrm{EOS}(h_1, s_1) \\ [p_2, \rho_2, a_2, q_2] = \mathrm{EOS}(h_2, s_2) \\ [p_3, \rho_3, a_3, q_3] = \mathrm{EOS}(h_3, s_3) \\ [p_4, \rho_4, a_4, q_4] = \mathrm{EOS}(h_4, s_4) \\ \end{gather}

The equations for conservation of mass are used in order to compute all the radii and blade heights at all stations. From the mass balance in the stator we find that:

H2H1=ρ1ρ2r1r2\frac{H_2}{H_1} = \frac{\rho_1}{\rho_2} \, \frac{r_1}{r_2}

from the definition of aspect ratio of the stator

H2r2=2ARS(1r1/r2)(1+H1/H2)1\frac{H_2}{r_2} = 2 AR_{\mathrm{S}}( 1 - r_1/r_2)(1 + H_1/H_2)^{-1}

Inserting these results into the definition of mass flow rate we find that

m˙=ρ2vmA2=ρ2vm2πr2H2m˙=ρ2v22πr22[ARS(1r1/r2)(1+H1/H2)1]r22=m˙4πvmρ2ARS1(1r1/r2)1(1+H1/H2)\begin{gather} \dot{m} = \rho_2 v_m A_2 = \rho_2 v_m 2\pi r_2 H_2 \\ \dot{m} = \rho_2 v_2 2\pi r_2^2 \left[AR_{\mathrm{S}} \, ( 1 - r_1/r_2)(1 + H_1/H_2)^{-1} \right] \\ r_2^2 = \frac{\dot{m}}{4\pi v_m \rho_2} AR_{\mathrm{S}}^{-1} \, ( 1 - r_1/r_2)^{-1}(1 + H_1/H_2) \end{gather}

Once r2r_2 is known, the radius at the other stations is given by:

r1=r2(r1/r2)r3=r2(r2/r3)1r4=r3(r3/r4)1\begin{gather} r_1 = r_2 \cdot (r_1 / r_2) \\ r_3 = r_2 \cdot (r_2 / r_3)^{-1} \\ r_4 = r_3 \cdot (r_3 / r_4)^{-1} \end{gather}

The blade height at each station is thus given by:

H1=m˙(2πr1ρ1vm)1H2=m˙(2πr2ρ2vm)1H3=m˙(2πr3ρ3vm)1H4=m˙(2πr4ρ4vm)1\begin{gather} H_1 = \dot{m} (2\pi r_1 \rho_1 v_m)^{-1} \\ H_2 = \dot{m} (2\pi r_2 \rho_2 v_m)^{-1} \\ H_3 = \dot{m} (2\pi r_3 \rho_3 v_m)^{-1} \\ H_4 = \dot{m} (2\pi r_4 \rho_4 v_m)^{-1} \\ \end{gather}

In order to compute the number of blades and openng of the stator blades we can use the Zweiffel criterion to determine the spacing-to-chord ratio

sSr2r1=0.5ZScos2 ⁣(α2)(tanα1tanα2)NS=π(r1+r2)/sSoS=sScos(α2+122πNS)\begin{gather} \frac{s_{\mathrm{S}}}{r_2- r_1} = \frac{0.5 Z_{\mathrm{S}}}{\cos^{2}\!\left( \alpha_{\mathrm{2}} \right) \left( \tan \alpha_{\mathrm{1}} - \tan \alpha_{\mathrm{2}} \right)} \\ N_{\mathrm{S}} = \pi (r_1 + r_2) / s_{\mathrm{S}} \\ o_{\mathrm{S}} = s_{\mathrm{S}} \cos(\alpha_2 + \tfrac{1}{2}\tfrac{2\pi}{N_{\mathrm{S}}}) \end{gather}

Similarly, for the rotor we have that:

sRr4r3=0.5ZRcos2 ⁣(β4)(tanβ3tanβ4)NR=π(r4+r3)/sRoR=sRcos(β4+122πNR)\begin{gather} \frac{s_{\mathrm{R}}}{r_4- r_3} = \frac{0.5 Z_{\mathrm{R}}}{\cos^{2}\!\left( \beta_{\mathrm{4}} \right) \left( \tan \beta_{\mathrm{3}} - \tan \beta_{\mathrm{4}} \right)} \\ N_{\mathrm{R}} = \pi (r_4 + r_3) / s_{\mathrm{R}} \\ o_{\mathrm{R}} = s_{\mathrm{R}} \cos(\beta_4 + \tfrac{1}{2}\tfrac{2\pi}{N_{\mathrm{R}}}) \end{gather}